Question

Solve the equation 2tan2xcos2x+√3=0 for -180<thetre>180

Answer 1

To solve the equation 2tan(2x)cos(2x) + √3 = 0 for -180° < θ < 180°, we can use substitution.

Let u = 2x, so the equation becomes:

2tan(u)cos(u) + √3 = 0

Using the identity tan(u) = sin(u)/cos(u), we can rewrite the equation as:

2(sin(u)/cos(u))cos(u) + √3 = 0

Simplifying, we have:

2sin(u) + √3cos(u) = 0

Now, we can use the identity sin(a + b) = sin(a)cos(b) + cos(a)sin(b).

Let's rewrite the equation using this identity:

2sin(u + π/3) = 0

Setting sin(u + π/3) = 0, we get:

u + π/3 = 0

u = -π/3

Substituting back in, we have:

2x = -π/3

x = -π/6

Therefore, the solution to the equation 2tan(2x)cos(2x) + √3 = 0 for -180° < θ < 180° is x = -π/6.